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 A227135 Partitions with parts repeated at most twice and repetition only allowed if first part has an even index (first index = 1). 3
 1, 1, 1, 2, 2, 4, 4, 6, 8, 10, 12, 17, 20, 25, 31, 39, 47, 58, 69, 85, 102, 123, 145, 175, 207, 246, 290, 343, 401, 473, 551, 646, 751, 875, 1012, 1177, 1358, 1570, 1807, 2083, 2389, 2746, 3140, 3597, 4106, 4690, 5337, 6082, 6907, 7848, 8895, 10085, 11404, 12902, 14561, 16438, 18520, 20864, 23460, 26385, 29619 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 FORMULA Conjecture: A227134(n) + A227135(n) = A182372(n) for n>=0, see comment in A182372. G.f.: 1/(1-x) + Sum_{n>=2} x^(A002620(n+2)-1) / Product_{k=1..n} (1-x^k), where A002620(n) = floor(n/2)*ceiling(n/2) forms the quarter-squares. - Paul D. Hanna, Jul 06 2013 a(n) ~ c * exp(Pi*sqrt(2*n/5)) / n^(3/4), where c = 0.1291995618069... - Vaclav Kotesovec, May 28 2018 EXAMPLE G.f.: 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 8*x^8 +... G.f.: 1/(1-x) + x^3/((1-x)*(1-x^2)) + x^5/((1-x)*(1-x^2)*(1-x^3)) + x^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + x^11/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) + x^15/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6))) +... There are a(13)=25 such partitions, displayed here as partitions into two sorts of parts (format P:S for sort:part) where the first sort is 1 and sorts oscillate: 01:  [ 1:1  2:0  2:1  3:0  5:1  ] 02:  [ 1:1  2:0  2:1  4:0  4:1  ] 03:  [ 1:1  2:0  2:1  8:0  ] 04:  [ 1:1  2:0  3:1  7:0  ] 05:  [ 1:1  2:0  4:1  6:0  ] 06:  [ 1:1  2:0 10:1  ] 07:  [ 1:1  3:0  3:1  6:0  ] 08:  [ 1:1  3:0  4:1  5:0  ] 09:  [ 1:1  3:0  9:1  ] 10:  [ 1:1  4:0  8:1  ] 11:  [ 1:1  5:0  7:1  ] 12:  [ 1:1  6:0  6:1  ] 13:  [ 1:1 12:0  ] 14:  [ 2:1  3:0  3:1  5:0  ] 15:  [ 2:1  3:0  8:1  ] 16:  [ 2:1  4:0  7:1  ] 17:  [ 2:1  5:0  6:1  ] 18:  [ 2:1 11:0  ] 19:  [ 3:1  4:0  6:1  ] 20:  [ 3:1  5:0  5:1  ] 21:  [ 3:1 10:0  ] 22:  [ 4:1  9:0  ] 23:  [ 5:1  8:0  ] 24:  [ 6:1  7:0  ] 25:  [13:1  ] MAPLE ## See A227134 # second Maple program: b:= proc(n, i, t) option remember; `if`(n=0, 1-t,       `if`(i*(i+1) add(b(n\$2, t), t=0..1): seq(a(n), n=0..60);  # Alois P. Heinz, Feb 15 2017 MATHEMATICA b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1 - t, If[i*(i + 1) < n, 0, Sum[ b[n - i*j, i - 1, Mod[t + j, 2]], {j, 0, Min[t + 1, n/i]}]]]; a[n_] := Sum[b[n, n, t], {t, 0, 1}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *) PROG (PARI) {A002620(n)=floor(n/2)*ceil(n/2)} {a(n)=polcoeff(1/(1-x+x*O(x^n)) + sum(m=2, sqrtint(4*n), x^(A002620(m+2)-1)/prod(k=1, m, 1-x^k+x*O(x^n))), n)} for(n=0, 60, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 06 2013 CROSSREFS Cf. A227134 (parts may repeat after odd index). Sequence in context: A057601 A294150 A087135 * A162417 A240012 A295261 Adjacent sequences:  A227132 A227133 A227134 * A227136 A227137 A227138 KEYWORD nonn AUTHOR Joerg Arndt, Jul 02 2013 STATUS approved

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Last modified November 14 12:36 EST 2018. Contains 317185 sequences. (Running on oeis4.)